Students’ Mental Acts of Anticipating in Solving Problems involving Algebraic Inequalities and Equations
Anticipating is the mental act of conceiving a certain expectation without performing a sequence of detailed operations to arrive at the expectation. This dissertation seeks to characterize students’ problem-solving in terms of two types of anticipating acts: (a) foreseeing an action, which refers to the act of conceiving an expectation that leads to an action, prior to performing the operations associated with the action, and (b) predicting a result, which refers to the act of conceiving an expectation for the result of an event without actually performing the operations associated with the event. Harel’s (in press) triad of determinants—mental act, ways of understanding, and ways of thinking—is used to analyze students’ acts of foreseeing and predicting.
This research has three objectives: (a) to categorize students’ ways of thinking associated with foreseeing and predicting, (b) to identify the relationships between these ways of thinking and students’ ways of understanding inequalities/equations, and (c) to explore the potential for advancing students’ ways of thinking associated with foreseeing/predicting. To accomplish these goals, fourteen 11th graders enrolled in various mathematics courses were interviewed. Four of them participated in one-on-one teaching interventions. Non-directive tasks were used to elicit students’ anticipatory behaviors.
In this study, five ways of thinking associated with foreseeing were identified: impulsive anticipation, tenacious anticipation, explorative anticipation, analytic anticipation, and interiorized anticipation. Three ways of thinking associated with predicting were identified: association-based prediction, comparison-based prediction, and coordination-based prediction. In addition, five ways of understanding inequalities/equations (I/E) were identified: I/E-as-a-signal-for-procedure, I/E-as-a-static-comparison, I/E-as-a-proposition, I/E-as-a-constraint, and I/E-as-a-comparison-of-functions. Students’ ways of thinking associated with foreseeing/predicting were found to be related to the quality of their solutions as well as the sophistication of their ways of understanding inequalities/equations.
One learner’s improvement was summarized in terms of the change in the sub context (Cobb, 1985) in which she operated, from manipulating symbols in the pre interview to reasoning with symbols in the post-interview. Her operating in the sub-context of working with numbers helped her to achieve this transition. This finding underscores the importance of using numbers as a platform for algebra students to explore algebraic expressions and symbolic structures.
Kien Hwa Lim. 2006. "Students’ Mental Acts of Anticipating in Solving Problems involving Algebraic Inequalities and Equations" The Selected Works of Kien Lim
Available at: http://works.bepress.com/kien_lim/1